Hölder regularity for non-divergence-form elliptic equations with discontinuous coefficients

Fazio, Giuseppe Di; Fanciullo, Maria Stella; Zamboni, Pietro

doi:10.1016/j.jmaa.2013.05.029

Cited by 28 publications

(47 citation statements)

References 11 publications

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“…These are the earliest papers about nondivergence-type equations with VMO coefficients. For divergence-type equations with VMO/BMO coefficients, similar results were obtained in [11,2], and later in [3,4]. On the other hand, in [25,26] Krylov gave a unified approach to investigating the L p -solvability of both divergence and nondivergence form parabolic and elliptic equations with coefficients that are BMO in the spatial…”

supporting

confidence: 61%

Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients

Dong¹,

Kim²

2011

SIAM J. Math. Anal.

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We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding W 1 p -solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, Arch. Ration. Mech. Anal., 196 (2010), pp. 25-70]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [Arch. Ration. Mech. Anal., 96 (1986), pp. 81-96] on gradient estimates for elasticity system Dα(A αβ (x 1 )D β u ) = f , which typically arises in homogenization of layered materials. We relax the condition on f from H k , k ≥ d/2, to Lp with p > d. Introduction.In this paper we prove the unique solvability of divergencetype fully coupled parabolic and elliptic systems in Sobolev spaces when the leading coefficients are in the class of variably partially bounded mean oscillation (BMO) functions. The distinguishing feature of variably partially BMO coefficients is that they are allowed to be very irregular with respect to one spatial direction, and the spatial direction needs to be determined only locally. Thus the systems studied in this paper may model deformations in composite media as fiber-reinforced materials (see, e.g., [8] and [28]).There are many papers concerning elliptic and parabolic equations/systems in Sobolev spaces with vanishing mean oscillation (VMO) or BMO-type coefficients. Chiarenza, Frasca, and Longo first proved the interior estimate for nondivergence form elliptic equations with VMO coefficients [6]. Then the solvability of elliptic and parabolic equations in Sobolev space was presented in [7] and [5]. These are the earliest papers about nondivergence-type equations with VMO coefficients. For divergence-type equations with VMO/BMO coefficients, similar results were obtained in [11,2], and later in [3,4]. On the other hand, in [25,26] Krylov gave a unified approach to investigating the L p -solvability of both divergence and nondivergence form parabolic and elliptic equations with coefficients that are BMO in the spatial

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supporting

confidence: 61%

Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients

Dong¹,

Kim²

2011

SIAM J. Math. Anal.

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“…An examination of the proofs in [3] and [10] shows that for all a ∈ A Φ the constant in (4.22) is uniformly bounded, with a bound, depending on Φ, D, d, λ, Λ. Therefore, according to the estimate (4.22), for each 0 < θ < 1, the solution u a satisfies the gradient condition GC(θ, M ) with M only depending on θ, D, d, λ, Λ, Φ, and f .…”

Section: Vmo Diffusion Coefficientsmentioning

confidence: 99%

Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

Bonito¹,

Cohen

DeVore

et al. 2017

SIAM J. Math. Anal.

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This paper considers the Dirichlet problem, where a is a scalar diffusion function. For a fixed f , we discuss under which conditions is a uniquely determined and when can a be stably recovered from the knowledge of u a . A first result is that whenever a ∈ H 1 (D), with 0 < λ ≤ a ≤ Λ on D, and f ∈ L ∞ (D) is strictly positive, thenMore generally, it is shown that the assumption a ∈ H 1 (D) can be weakened to a ∈ H s (D), for certain s < 1, at the expense of lowering the exponent 1/6 to a value that depends on s.

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“…We mention some recent results on the second order elliptic equations and systems with coefficients in VMO due to Di Fazio [23], Caffarelli and Peral [12], Stroffolini [68], Guidetti [32], Auscher and Qafsaouti [9], for higher order equations see also Palagachev and Softova [62]. Recently Dindoš et al obtained interesting facts concerning the solvability of the Dirichlet problem with BMO boundary data for second order scalar uniformly elliptic equations with real coefficients on a very general class of domains, including Lipschitz and polyhedral domains [24].…”

Section: (N−1)mentioning

confidence: 99%

Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

Maz′ya

Shaposhnikova

2011

Bull. Math. Sci.

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Hölder regularity for non-divergence-form elliptic equations with discontinuous coefficients

Abstract: In this note we study the global regularity in the Morrey spaces L p,λ for the second derivatives for the strong solutions of non variational elliptic equations.

Cited by 28 publications

References 11 publications

Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients

Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients

Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

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